Topological groups with a bc-base

Approximately 10 years ago, Zambakhidze asked whether every non-zero-dimensional topological group with a bc-base is locally compact. Below we show that the small inductive dimension ind of any non-locally compact group with such a base doesn't exceed 1. We prove, however, that a σ-compact non-locally compact topological group with a bc -base is zero-dimensional. Two more results in this paper are worth mentioning: 1) if the free topological group F(X)F(X) of a Tychonoff space X has a bc -base, then ind(X)≤0ind(X)≤0, and 2) a topological group G has a bc-base if and only if G can be compactified by a zero-dimensional remainder.